Nuclear lineshape

Binsch G 1969 A unified theory of exchange effects on nuclear magnetic resonance lineshapes J. Am. Chem. Soc. 91 1304-9... 

A unique situation is encountered if Fe-M6ssbauer spectroscopy is applied for the study of spin-state transitions in iron complexes. The half-life of the excited state of the Fe nucleus involved in the Mossbauer experiment is tj/2 = 0.977 X 10 s which is related to the decay constant k by tj/2 = ln2/fe. The lifetime t = l//c is therefore = 1.410 x 10 s which value is just at the centre of the range estimated for the spin-state lifetime Tl = I/Zclh- Thus both the situations discussed above are expected to appear under suitable conditions in the Mossbauer spectra. The quantity of importance is here the nuclear Larmor precession frequency co . If the spin-state lifetime Tl = 1/feLH is long relative to the nuclear precession time l/co , i.e. Tl > l/o) , individual and sharp resonance lines for the two spin states are observed. On the other hand, if the spin-state lifetime is short and thus < l/o) , averaged spectra with intermediate values of quadrupole splitting A q and isomer shift 5 are found. For the intermediate case where Tl 1/cl , broadened and asymmetric resonance lines are obtained. These may be the subject of a lineshape analysis that will eventually produce values of rate constants for the dynamic spin-state inter-conversion process. The rate constants extracted from the spectra will be necessarily of the order of 10 -10 s"F... 

In molecularly rigid systems, the direct (through-space) dipole-dipole interaction between nuclear spins I = fe is normally the dominant source of broadening of the NMR lineshape. For a pair of similar nuclear spins i and j in a magnetic field Hq the dipolar splitting in their spectrum is given by... 

The great utility of moments is that, although the lineshape cannot be calculated analytically for an arbitrary configuration of nuclear spins, any moment may in principle be calculated to arbitrary precision from first principles (10). In practice, only the lowest moments are calculable because of computer time and precision constraints. In particular, the second moment M2 is the lowest moment containing spacial information... 

An interesting new approach has been made possible by multiple quantum 27A1 MAS (MQMAS) NMR investigations of zeolites. Due to its nuclear electric quad-rupole moment, 27A1 exhibits broad lineshapes in distorted structural environments. 

We describe in some detail the techniques of nuclear magnetic resonance which are used for studying alumina-supported platinum catalysts. In particular, we describe the spin-echo technique from which the Pt lineshape can be obtained. We also discuss spin echo double resonance between surface Pt and chemisorbed molecules and show how the NMR resonance of the surface Pt can be separately studied. We present examples of experimental data and discuss their interpretation. 

In a very broad line, then, the size of the spin echo is proportional to the number of nuclear moments near go. If we change Hq, we change h by the same amount and thus sample nuclear moments at a different local field. By measuring the size of the spin echo as a function of Hq, we obtain the NMR lineshape, i.e., in this case, the distribution of the local field among the nuclear spins in the sample. 

In a sample of bulk Pt metal, all of the nuclei have the same interaction with the conduction electrons and thus see the same local field. The resulting NMR line is quite narrow. However, in our samples of small Pt particles, many of the nuclei are near a surface where the state of the conduction electron is disturbed. This tends to reduce the Knight shift for these nuclei. Since the Pt particles in our samples are of many different sizes and shapes, this reduction in the Knight shift is not the same for every nuclear spin near a surface. Thus, we obtain a broad "smear" of Knight shifts resulting in the lineshapes of Figure 5. 

Motions with rates of the order of the nuclear spin interaction anisotropy can be assessed via lineshape analysis. These are generally motions of intermediate rates, a few kHz to tens of kHz for chemical shift and dipolar interactions, higher for quadrupolar interactions. 

Fig. 2. (a) The free induction decay, G(t) for 19F in a single crystal of CaFi for B0 along [1,0,0]. The experimental points are given by circles and crosses from the CW and pulse measurements, respectively, and the theoretical curve is that of Eq. (14), corresponding to an exponential decay multiplied by a sine function. Note that F(t) is equivalent to G(t) in the present notation. Reproduced with permission from A. Abragam, The Principles of Nuclear Magnetism, p. 121, Oxford University Press, London, 1961. (b) The lineshape in the frequency domain corresponding to the Fourier transform of the theoretical curve. 

Fig. 3. Calculated NMR lineshapes for equally populated two-site exchange as a function of the dimensionless parameter a = nf rA. The abscissa is the dimensionless relative offset parameter, x = A/// (see Eq. (18)). (a) a = 4 (b) a = 2 (c)a=l (d) a = l/ /2 (e) a = 0.5 (f) a = 0.2. Spectra (a) and (f) are near the slow and fast exchange limits, respectively. Reproduced with permission from R. K. Harris, Nuclear Magnetic Resonance Spectroscopy A Physicochemical View, p. 124, Longman Scientific and Technical, Harlow, 1986.

However, the NMR properties of solid-phase methane are very complex, due to subtle effects associated with the permutation symmetry of the nuclear spin set and molecular rotational tunnelling.55 Nuclear spin states ltotai = 0 (irred. repr. E), 1 (T) and 2 (A) are observed. The situation is made more complicated since, as the solids are cooled and the individual molecules go from rotation to oscillation, several crystal phases become available, and slow transitions between them take place. Much work has been done in the last century on this problem, including use of deuterated versions of methane for example see Refs. 56-59. Much detail has emerged from NMR lineshape analysis and relaxation time measurements, and kinetic studies. For example, the second moment of the 13C resonance is found to be caused by intermolecular proton-carbon spin-spin interaction.60 Thus proton inequivalence within the methane molecules is created. 

Below that temperature this thermally activated motion is frozen out. There is a characteristic change in the solid-state lineshape that clearly demonstrates the effect. Nuclear magnetic resonance can in such cases provide important information about molecular structure and dynamics in solids. 

A special case occurs if the solid contains nuclear spin pairs that are sufficiently far apart from each other such that intrapair coupling dominates interpair coupling. In this situation, instead of a featureless broad lineshape, we observe a well-defined powder pattern known as a Pake doublet (Figure 15.15). Since the dipolar coupling between two nuclei... 

In the solid state, all interactions of nuclear spins exhibit anisotropic behavior, leading to complex spectra and broad lineshapes, the latter being the source for extremely short T2 and T Because of reduced molecular motion, 7) typically becomes very long. 

The analytical expression for the perturbation function is known, and this is fitted to the experimentally determined perturbation function. This leads to the determination of not only the transition frequencies described above, but also of other parameters related to amphtude and lineshape of the signal, as described in the following text. For a spin 5/2 intermediate nuclear state, the data analysis leads to the determination of >i, >2, and m3. Usually independent parameters (recall that only two of these frequencies are independent). There are various traditions in terms of which parameters to report, but qualitatively one refers to the NQI strength vq = eq Q/h (or mg = mp = 2n/(AI(2I - l))vg). 

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